arXiv subjects

Brent Cody

Publications and source records attributed to Brent Cody.

At least 19 recordsLinked to original sources

arXiv2026

Metric general position extensions of classical graph invariants and perfection

We introduce a two-parameter framework that refines several classical graph invariants by imposing higher-order constraints along bounded-length geodesics. For integers $k,d\ge1$, a vertex set is called $k,d$-independent if every shortest path of length at most $d$ contains fewer than $k$ vertices of the set, giving rise to corresponding $k,d$-independence, chromatic, clique, and domination invariants. We develop a general framework for these parameters by associating each graph with a $k$-uniform hypergraph that encodes its geodesic structure. We then establish basic bounds and monotonicity properties, and introduce a notion of $k,d$-perfection extending the classical theory of perfect graphs. Exact formulas are obtained for the $k,d$-chromatic number of paths and cycles. In particular, all paths are $k,d$-perfect for all parameters, while cycles admit a complete classification of $k,d$-perfection that recovers the classical case when $k=2$ and exhibits new periodic and finite-exception behavior for $k\ge3$. We further investigate the interaction between $k,d$-invariants and graph powers, showing that while the $k=2$ case reduces to graph powers in a straightforward way, substantially different behavior arises for higher values of $k$, even for powers of paths.

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arXiv2025

Two-cardinal ideal operators and indescribability

A well-known version of Rowbottom's theorem for supercompactness ultrafilters leads naturally to notions of two-cardinal Ramseyness and corresponding normal ideals introduced herein. Generalizing results of Baumgartner [7, 8], Feng [22] and the first author [16, 17], we study the hierarchies associated with a particular version of two-cardinal Ramseyness and a strong version of two-cardinal ineffability, as well as the relationships between these hierarchies and a natural notion of transfinite two-cardinal indescribability.

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arXiv2025

The Wiener index of vertex colorings

The Wiener index of a vertex coloring of a graph is defined to be the sum of all pairwise geodesic distances between vertices of the same color. We provide characterizations of vertex colorings of paths and cycles whose Wiener index is as large as possible over various natural collections. Along the way we establish a connection between the majorization order on tuples of integers and the Wiener index of vertex colorings on paths and cycles.

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arXiv2025

Sets of vertices with extremal energy

We define various notions of energy of a set of vertices in a graph, which generalize two of the most widely studied graphical indices: the Wiener index and the Harary index. We provide a new proof of a result due to Douthett and Krantz, which says that for cycles, the sets of vertices which have minimal energy among all sets of the same size are precisely the maximally even sets, as defined in Clough and Douthett's work on music theory. Generalizing a theorem of Clough and Douthett, we prove that a finite, simple, connected graph is distance degree regular if and only if whenever a set of vertices has minimal energy, its complement also has minimal energy. We also provide several characterizations of sets of vertices in finite paths and cycles for which the sum of all pairwise distances between vertices in the set is maximal among all sets of the same size.

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arXiv2024

The k-general d-position problem for graphs

A set of vertices of a graph is said to be in general position if no three vertices from the set lie on a common geodesic. Recently Klavžar, Rall and Yero generalized this notion by defining a set of vertices to be in general $d$-position if no three vertices from the set lie on a common geodesic of length at most $d$. We generalize this notion further by defining a set of vertices to be in $k$-general $d$-position if no $k$ vertices of the set lie on a common geodesic of length at most $d$. The $k$-general $d$-position number of a graph is the largest cardinality of a $k$-general $d$-position set. We provide upper and lower bounds on the $k$-general $d$-position number of graphs in terms of the $k$-general $d$-position number of certain kinds of subgraphs. We compute the $k$-general $d$-position number of finite paths and cycles. Along the way we establish that the maximally even subsets of cycles, which were introduced in Clough and Douthett's work on music theory, provide the largest possible $k$-general $d$-position sets in $n$-cycles. We generalize Klavžar and Manuel's notion of monotone-geodesic labeling to that of $k$-monotone-geodesic labeling in order to calculate the $k$-general $d$-position number of the infinite two-dimensional grid. We also prove a formula for the $k$-general $d$-position number of certain thin finite grids, providing a partial answer to a question asked by Klavžar, Rall and Yero.

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arXiv2024

The Music and Mathematics of Maximal Evenness in Graphs

We use the concept of electric potential energy from physics, the mathematical field of graph theory, and the notion of majorization to study maximal evenness in a broader mathematical context than what was previously possible, so that we can go beyond the well-known one-dimensional maximally even sets into higher dimensional and more geometrically complex territory. We investigate musical connections between certain generalizations of maximally even sets, one of the oldest Puerto Rican musical traditions of African origin called bomba, and with certain scales ranging from the familiar to the esoteric.

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arXiv2024

Two-cardinal derived topologies, indescribability and Ramseyness

We introduce a natural two-cardinal version of Bagaria's sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Peter Holy and Philip White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

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arXiv2023

Sparse analytic systems

Erdős \cite{MR168482} proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal{F}$ of (real or complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{F} \big\}$ is countable for every $x$. We strengthen Erdős' result by proving that CH is equivalent to the existence of what we call \emph{sparse analytic systems} of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim$ on $\mathbb{R}$ such that any "analytic-anonymous" attempt to predict the map $x \mapsto [x]_\sim$ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman \cite{MR3552748}.

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arXiv2022

Higher indescribability and derived topologies

We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{κ^+,κ^+}$-indescribability and $Π^1_ξ$-indescribability of a cardinal $κ$ for all $ξ<κ^+$. In this context, universal $Π^1_ξ$ formulas exist, there is a normal ideal associated to $Π^1_ξ$-indescribability and the notions of $Π^1_ξ$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal $μ$, we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's \cite{MR3894041} sequence $langleτ_ξ:ξ<μ\rangle$ of derived topologies on $μ$ to $\langleτ_ξ:ξ<μ^+\rangle$. Finally, we prove that for all $ξ<μ^+$, if there is a stationary set of $α<μ$ that have a high enough degree of indescribability, then there are stationarily-many $α<μ$ that are nonisolated points in the space $(μ,τ_{ξ+1})$.

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arXiv2022

Ideal operators and higher indescribability

We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al. and by the first author.

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arXiv2021

Adding a non-reflecting weakly compact set

For $n<ω$, we say that the $Π^1_n$-reflection principle holds at $κ$ and write $\text{Refl}_n(κ)$ if and only if $κ$ is a $Π^1_n$-indescribable cardinal and every $Π^1_n$-indescribable subset of $κ$ has a $Π^1_n$-indescribable proper initial segment. The $Π^1_n$-reflection principle $\text{Refl}_n(κ)$ generalizes a certain stationary reflection principle and implies that $κ$ is $Π^1_n$-indescribable of order $ω$. We define a forcing which shows that the converse of this implication can be false in the case $n=1$. Moreover, we prove that if $κ$ is $(α+1)$-weakly compact where $α<κ^+$, then there is a forcing extension in which there is a weakly compact set $W\subseteqκ$ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $κ$ remains $(α+1)$-weakly compact. Additionally, we prove a resurrection result for the $Π^1_1$-reflection principle.

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arXiv2021

Large cardinal ideals

Building on work of Holy, Lücke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of several well-known large cardinal ideals, including the Ramsey ideal, in terms of generic elementary embeddings; we also point out some seemingly inherent differences between small embedding and generic embedding characterizations of subtle cardinals. Additionally, we present a simple and uniform proof which shows that, when $κ$ is weakly compact, many large cardinal ideals on $κ$ are nowhere $κ$-saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and $Π^1_1$-indescribable ideals on $P_κλ$, and we close with a list of open questions.

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arXiv2020

Forcing a $\square(κ)$-like principle to hold at a weakly compact cardinal

Hellsten \cite{MR2026390} proved that when $κ$ is $Π^1_n$-indescribable, the \emph{$n$-club} subsets of $κ$ provide a filter base for the $Π^1_n$-indescribability ideal, and hence can also be used to give a characterization of $Π^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteqκ$. By replacing clubs with $n$-clubs in the definition of $\Box(κ)$, one obtains a $\Box(κ)$-like principle $\Box_n(κ)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(κ)$ is consistent with the $Π^1_n$-indescribability of $κ$ but inconsistent with the $Π^1_{n+1}$-indescribability of $κ$. By generalizing the standard forcing to add a $\Box(κ)$-sequence, we show that if $κ$ is $κ^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $κ$ remains $κ^+$-weakly compact and $\Box_1(κ)$ holds. If $κ$ is $Π^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $κ$ is $κ^+$-weakly compact, $\Box_1(κ)$ holds and every weakly compact subset of $κ$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $Π^1_2$-indescribable cardinal, it is consistent that $κ$ is $κ^+$-weakly compact, every weakly compact subset of $κ$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $κ$ such that there is no $β<κ$ for which both $S^0\capβ$ and $S^1\capβ$ are weakly compact.

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arXiv2020

A refinement of the Ramsey hierarchy via indescribability

A subset $S$ of a cardinal $κ$ is Ramsey if for every function $f:[S]^{<ω}\to κ$ with $f(a)<\min a$ for all $a\in[S]^{<ω}$, there is a set $H\subseteq S$ of cardinality $κ$ which is \emph{homogeneous} for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<ω$. Baumgartner proved \cite{MR0384553} that if $κ$ is a Ramsey cardinal, then the collection of non-Ramsey subsets of $κ$ is a normal ideal on $κ$. Sharpe and Welch \cite{MR2817562}, and independently Bagaria \cite{MR3894041}, extended the notion of $Π^1_n$-indescribability where $n<ω$ to that of $Π^1_ξ$-indescribability where $ξ\geqω$. We study large cardinal properties and ideals which result from Ramseyness properties in which homogeneous sets are demanded to be $Π^1_ξ$-indescribable. By iterating Feng's Ramsey operator \cite{MR1077260} on the various $Π^1_ξ$-indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng's original Ramsey hierarchy. We also show that, given any ordinals $β_0,β_1<κ$ the increasing chains of ideals obtained by iterating the Ramsey operator on the $Π^1_{β_0}$-indescribability ideal and the $Π^1_{β_1}$-indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $Π^1_ξ$-indescribability and Ramseyness.

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arXiv2020

Characterizations of the weakly compact ideal on $P_κλ$

Hellsten \cite{MR2026390} gave a characterization of $Π^1_n$-indescribable subsets of a $Π^1_n$-indescribable cardinal in terms of a natural filter base: when $κ$ is a $Π^1_n$-indescribable cardinal, a set $S\subseteqκ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq κ$. We generalize Hellsten's characterization to $Π^1_n$-indescribable subsets of $P_κλ$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $Π^1_0$-indescribability ideal on $P_κλ$ equals the minimal \emph{strongly} normal ideal $\text{NSS}_{κ,λ}$ on $P_κλ$, and is not equal to $\text{NS}_{κ,λ}$ as may be expected, we formulate a notion of $n$-club subset of $P_κλ$ and prove that a set $S\subseteq P_κλ$ is $Π^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq P_κλ$. We also prove that elementary embeddings considered by Schanker \cite{MR2989393} witnessing \emph{near supercompactness} lead to the definition of a normal ideal on $P_κλ$, and indeed, this ideal is equal to Baumgartner's ideal of non--$Π^1_1$-indescribable subsets of $P_κλ$. Additionally, as applications of these results we answer a question of Cox-Lücke \cite{MR3620068} about $\mathcal{F}$-layered posets, provide a characterization of $Π^m_n$-indescribable subsets of $P_κλ$ in terms of generic elementary embeddings, prove several results involving a two-cardinal weakly compact diamond principle and observe that a result of Pereira \cite{MR3640048} yeilds the consistency of the existence of a $(κ,κ^+)$-semimorasses $μ\subseteq P_κκ^+$ which is $Π^1_n$-indescribable for all $n<ω$.

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arXiv2019

Rigid ideals

An ideal $I$ on a cardinal $κ$ is called \emph{rigid} if all automorphisms of $P(κ)/I$ are trivial. An ideal is called \emph{$μ$-minimal} if whenever $G\subseteq P(κ)/I$ is generic and $X\in P(μ)^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $μ$-minimal ideal on $μ^+$, where $μ$ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on $μ^+$, where $μ$ is an \emph{uncountable} regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case $μ=ω$, we show that the existence of a rigid \emph{presaturated} ideal on $ω_1$ is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a \emph{precipitous} rigid ideal on $μ^+$ where $μ$ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.

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arXiv2017

The weakly compact reflection principle need not imply a high order of weak compactness

The weakly compact reflection principle $\text{Refl}_{\text{wc}}(κ)$ states that $κ$ is a weakly compact cardinal and every weakly compact subset of $κ$ has a weakly compact proper initial segment. The weakly compact reflection principle at $κ$ implies that $κ$ is an $ω$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $κ$ is $(ω+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $κ$ then there is a forcing extension preserving this in which $κ$ is the least $ω$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $κ$ is a regular cardinal then in any forcing extension by $κ$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $κ$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $κ$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.

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arXiv2015

Indestructibility of generically strong cardinals

Foreman proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $ω_1$ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie in his PhD Thesis, Universitat Munster, 2010). As an application we prove that if $ω_1$ is generically strong, then it remains so after adding any number of Cohen subsets of $ω_1$; however many other $ω_1$-closed posets---such as $\text{Col}(ω_1, ω_2)$---can destroy the generic strength of $ω_1$. This generalizes some results of Gitik-Shelah about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than $ω_1$.

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